Journal of Dynamics and Differential Equations

, Volume 6, Issue 1, pp 53-69

First online:

Completely integrable bi-Hamiltonian systems

  • Rui L. FernandesAffiliated withSchool of Mathematics, University of MinnesotaDepartamento de Matemática, Instituto Superior Técnico

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We study the geometry of completely integrable bi-Hamiltonian systems and, in particular, the existence of a bi-Hamiltonian structure for a completely integrable Hamiltonian system. We show that under some natural hypothesis, such a structure exists in a neighborhood of an invariant torus if, and only if, the graph of the Hamiltonian function is a hypersurface of translation, relative to the affine structure determined by the action variables. This generalizes a result of Brouzet for dimension four.

Key words

Bi-Hamiltonian system completely integrable system